Efficient Ohmic contacts and built-in atomic sublayer protection in MoSi2N4 and WSi2N4 monolayers

Metal contacts to two-dimensional (2D) semiconductors are often plagued by the strong Fermi level pinning (FLP) effect which reduces the tunability of the Schottky barrier height (SBH) and degrades the performance of 2D semiconductor devices. Here, we show that MoSi2N4 and WSi2N4 monolayers—an emerging 2D semiconductor family with exceptional physical properties—exhibit strongly suppressed FLP and wide-range tunable SBH. An exceptionally large SBH slope parameter of S ≈ 0.7 is obtained which outperforms the vast majority of other 2D semiconductors. Such intriguing behavior arises from the septuple-layered morphology of MoSi2N4 and WSi2N4 monolayers in which the semiconducting electronic states are protected by the outlying Si–N sublayer. We identify Ti, Sc, and Ni as highly efficient Ohmic contacts to MoSi2N4 and WSi2N4 with zero interface tunneling barrier. Our findings reveal the potential of MoSi2N4 and WSi2N4 as a practical platform for designing high-performance and energy-efficient 2D semiconductor electronic devices.


INTRODUCTION
Electrical contacts between metals and semiconductors are ubiquitous in modern electronic and optoelectronic devices. An interfacial potential barrier, known as the Schottky barrier (SB), is commonly formed at the metal/semiconductor interface. In electronics and optoelectronics applications, the presence of a sizable Schottky barrier height (SBH), typically larger than a few k B T, can severely impede the charge injection efficiency 1 . As the SBH is intimately linked to the contact resistance at the metal/semiconductor contact 2,3 , reducing the SBH at the metal/semiconductor contact has become one of the key challenges towards energy-efficient and high-speed semiconductor devices.
In a metal/semiconductor contact, SBH arises from the mismatch between the metal work function (W M ) and the semiconductor electron affinity E ea (n-type Schottky contact) or ionization potential E ip (p-type Schottky contact). The SBH across a metal/semiconductor contact can be phenomenologically captured by the modified Schottky-Mott (SM) rule [4][5][6][7] , where the subscript "e" and "h" denote n-and p-type contacts, respectively, c e(h) is a material-and contact-dependent term 7 , and Φ Be and Φ Bh is the electron-type and hole-type SBH, respectively. Here, the slope parameter, S e(h) , is defined as which is an important phenomenological parameter widely used in characterizing the deviation of the SBH from the ideal SM limit (S = 1). In realistic metal/semiconductor contacts, S ≪ 1 due to the presence of multiple nonideal factors, such as the formation of metal-induced gap states, defect-induced gap states, mid-gap states, and interface dipole, as well as the modifications of the electronic band structures of the semiconductor when contacted by metals 6,7 . In this case, the SBH is pinned to a narrow range of value-an adverse effect commonly known as the Fermi level pinning (FLP).
In the few-atom-thick limit, two-dimensional (2D) semiconductors 8 , such as MoS 2 and WS 2 9 , continue to be plagued by FLP 10 . Although 2D semiconductors and their van der Waals (VDW) heterostructures has shown great promises in low-power electronics 11 , optoelectronics 12 , and neuromorphic applications 13 , the lack of wide-range tunable SBH-commonplace in most 2D semiconductors-has severely impeded the development of high-performance nanodevices. Particularly, the metallization of a 2D semiconductor by the contacting metal often substantially alters the electronic structures of the heterostructure via the generation of mid-gap states 14,15 , causing strong FLP effect that leads to a poor SBH tunability 15 . For the vast majority of 2D semiconductors, S is typically less than 0.4 as predicted by density functional theory (DFT) calculations, and are even lower in experimental measurements due to the inevitable presence of defects at the contact interface. To resurrect a wide-range tunable SBH in 2D semiconductors, atomically sharp VDW metal/2D-semiconductor electrical contacts have been proposed 16 . Such VDW-type contacts harness the weak VDW interfacial coupling to reduce the metal/ semiconductor interactions, yielding an S ≈ 1 that approaches 1 Science, Mathematics and Technology, Singapore University of Technology and Design, Singapore, Singapore. 2 [17][18][19] . However, as VDW-type contact often involves complex fabrication techniques, a 2D semiconductor class that can inherently achieve substantial Fermi level unpinning without necessarily relying on the VDW contact paradigm remains elusive thus far.
In this work, we perform a first-principle DFT investigation to study the metal contact properties in the recently discovered synthetic 2D monolayers, MoSi 2 N 4 and WSi 2 N 4 20,21 . We focus on face-type contact 1 as it is widely employed in both the experimental design of 2D nanodevices 18,19,[22][23][24][25][26] and the computational contact simulation studies 15,17,[27][28][29] . The recent demonstrations of ultralow contact resistance and the high onstate current in face-contacted 2D semiconductor 18,19,22 also reveal the technological importance of face-type contact towards the development of high-performance 2D nanodevices 30 . We show that, in contrary to the common knowledge that 2D semiconductors are prone to strong FLP, MoSi 2 N 4 , and WSi 2 N 4 monolayers exhibit strongly suppressed FLP and excellent SBH tunability without relying on the VDW-type contact engineering. The SBH is widely tunable in MoSi 2 N 4 and WSi 2 N 4 monolayers, reaching an exceptionally high slope parameter of S = 0.69 and S = 0.77, respectively-a value much larger than other commonly studied 2D semiconductors. The FLP suppression originates from the unusual morphology and the electronic properties of MoSi 2 N 4 and WSi 2 N 4 monolayers, in which the semiconducting states residing in Mo-N or W-N inner core-layer are protected by the outlying Si-N atomic layers-an intriguing mechanism not found in other classes of 2D semiconductors. Our results reveal MoSi 2 N 4 and WSi 2 N 4 monolayer as an unusual 2D semiconductor class with built-in atomic layer protection, thus opening up an alternative and complementary route to the VDW contact paradigm towards efficient SBH tuning and high-performance electrical contact engineering.
In general, the metal contacts to MoSi 2 N 4 and WSi 2 N 4 cover a large variety of contact types, including n-type Schottky contact, p-type Schottky contact, and Ohmic contact with zero SBH. The interlayer distance between the metal and 2D monolayers is less than 3 Å for Cu, Pd, Ti, and Ni (see Supplementary Tables 1 and 2), suggesting the prevalence of non-VDW-type metal contacts. In the following, we shall take Au and Ti metal contacts as illustrative examples of Schottky (Fig. 2) and Ohmic contacts (Fig. 3), We further calculate the spatial charge density distribution around the VBM and the CBM, as well as the states between the VBM and the CBM. Intriguingly, the semiconducting electronic states are embedded deeply within inner Mo-N [Figs. 2c and 3c] and W-N core layer [Figs. 2f and 3f], while the mid-gap states between the VBM and CBM are mostly located in the metals and sparsely distributed in the outlying Si-N layers. Such metal-induced mid-gap states decay rapidly without penetrating into the semiconducting inner core layer, thus leading to a muchweakened FLP. We also calculate the differential charge density (Δρ) which reveals a significant charge redistribution across the metal/semiconductor contact interface [Figs. 2c, f and 3c, f]. As Δρ across the contact interface is asymmetrical, the formation of interface dipole is expected to modify the S parameter from the ideal SM limit (see Supplementary Fig. 4 for the calculated Δρ of other metal contacts).
The effective electrostatic potential profile, V eff , across the metal/semiconductor contact is shown in the left-most panels of Fig. 2c, f for Au contact, and Fig. 3c, f for Ti contact. For contacts with V eff higher than the Fermi level at the metal/semiconductor interface, the V eff forms an interfacial tunneling potential barrier (denoted as Φ TB in Figs. 2 and 3), which significantly impedes the charge injection efficiency. In the case of Au contact, the presence of a sizable barrier of Φ TB ≈ 3.13 eV [Fig. 2c, f] reveals a muchreduced electron transmission across the metal/semiconductor contact. In contrast, the absence of Φ TB in the case of Ti contact [ Fig. 3c, f] reveals the potential of Ti as an efficient electrode to MoSi 2 N 4 and WSi 2 N 4 with a large electron transmission probability. The Φ TB and the corresponding electron transmission probability of various metal contacts to MoSi 2 N 4 and WSi 2 N 4 will be discussed in detail below.
The absence of mid-gap states in the inner core layer and the finite charge transfer at the outlying Si-N layers reveals that the metal/semiconductor interaction affects mostly the outlying Si-N layers, predominantly via charge redistribution, without penetrating semiconducting Mo-N and W-N inner cores. Here the Si-N layer serves as an encapsulating layer and plays a vital role in b The projected electronic band structure and the partial density of states of Au/MoSi 2 N 4 contact. The panels in c from left to right, show the spatial charge density distribution of VBM states, mid-gap states between VBM and CBM, and the CBM states, the differential charge density, and the electrostatic potential profile across the heterostructures. d-f Same as a, b, and c, respectively, for Au/WSi 2 N 4 contact. The interlayer distance is 3.13 Å and the minimum distance between Au and N atoms is 3.54 Å.
preserving the semiconducting characteristics of MoSi 2 N 4 and WSi 2 N 4 monolayers. To verify the protective effect of Si-N layers 53 , we simulate a close contact type of Au/MoSi 2 N 4 and Au/WSi 2 N 4 heterostructures by forcing the interlayer distance to 1.5 Å, compared to the fully relaxed value of about 3.13 Å (see Supplementary Fig. 3). The semiconducting band structures are well-preserved at this close-contact limit, thus confirming the robustness of the semiconducting states residing in the MoSi 2 N 4 and WSi 2 N 4 monolayers and the resilience against mid-gap states formation. Such unusual behavior, enabled by the septuplelayered morphology of MoSi 2 N 4 and WSi 2 N 4 monolayer, is not found in other commonly studied 2D semiconductors, such as transition metal dichalcogenides (TMDCs) and black phosphorus, in which an external insertion layer is often required to unpin the Fermi level [54][55][56] . We further perform a comparison with monolayer MoS 2 by calculating the charge density distribution 39 and the electronic band structures of Sc/MoSi 2 N 4 and the Sc/MoS 2 contacts in Fig. 4. The charge density plots in Fig. 4a, c shows the presence of a large amount of electrons from Sc atoms overlapping strongly with the MoS 2 layer [ Fig. 4c] while the MoSi 2 N 4 layer remains well-isolated from the electrons of Sc atoms [ Fig. 4a]. Such electron overlap in Sc/MoS 2 directly leads to strong metal-semiconductor hybridization as evident from the band structure and the PDOS plot in Fig. 4d, which is in strong contrast to the case of Sc/MoSi 2 N 4 in Fig. 4b where the electronic states around CBM and VBM remain intact.
As the CBM and VBM electronic states contribute dominantly to the carrier transport, the suppression of FLP due to the atomiclayer protection as discussed above can also be understood as the consequence of having a transport channel that is spatially wellseparated from the influence of the contacting metal atoms. In the case of MoS 2 -a triple-layered structure which is morphologically much thinner than the septuple-layered MoSi 2 N 4 , the transport channel residing around the Mo atoms is in closer proximity with the contacting metal and hence the presence of stronger hybridization leads to a more severe FLP effect.
Interfacial tunneling potential barrier and Schottky barrier As discussed above, the formation of a tunneling potential barrier, Φ TB , at the metal/semiconductor gap impedes the charge injection efficiency 52 . The Φ TB and the barrier width, d TB can be determined from the effective electrostatic potential across the metal/semiconductor interface (see Supplementary Fig. 5). For both MoSi 2 N 4 and WSi 2 N 4 monolayers, Φ TB ranges between 2.8 and over 5.1 eV, and the thickness d TB ranges between 1.1 and 2 Å (see Supplementary Tables 1 and 2). The corresponding electron tunneling probability across the interface can be then calculated as, where m e is the free electron mass. The T ðΦ TB ; d TB Þ calculated via Eq. (4) for various metal contacts are shown in Fig. 5a. For Sc, Ti, and Ni, the tunneling probability reaches 100% as Φ TB = 0. Particularly for Sc and Ti, both Φ TB and Φ B are zero in both MoSi 2 N 4 and WSi 2 N 4 monolayers-an indication of good Ohmic contacts with high charge injection efficiency. In contrast, the b The projected electronic band structure and the partial density of states of Ti/MoSi 2 N 4 contact. The panels in c from left to right, show the spatial charge density distribution of VBM states, mid-gap states between VBM and CBM, and the CBM states, the differential charge density, and the electrostatic potential profile across the heterostructures. d-f Same as a, b, and c, respectively, for Ti/WSi 2 N 4 contact. The interlayer distance is 1.78 Å and the minimum distance between Ti and N atoms is 2.45 Å.
VDW-type contacts, such as graphene and NbS 2 with interlayer distances greater than 3 Å, exhibit very low T of less than 2%, which suggests a low electron transparency in such VDW-type interfaces. Apart from tunneling probability, the extracted d TB and Φ TB can also be used to estimate the tunneling-specific resistivity (ρ t ) 22 . Based on Simmons tunneling injection model 57 , ρ t can be obtained as, where α = 1 under the square-barrier approximation. For Ti and Sc contacts, ρ t ≈ 0 because of the absence of the interfacial tunneling barrier. For other metal contacts, the calculated ρ t lies in the typical range of 10 −9 Ωcm 2 which is comparable to that of the recently reported Bi/MoS 2 contact of ultralow contact resistance 22 . The calculated ρ t is listed in Supplementary Table 3 which can be readily employed for contact resistance calculation and for the analysis of device transport measurements 22 .
We now examine the SBHs of various metal contacts. As the semiconducting band structures of MoSi 2 N 4 and WSi 2 N 4 , especially the CBM and VBM states, are well preserved, the electron and hole SBH can be determined from the energy differences between the CBM and the Fermi level (ε F ) of the metal/ semiconductor heterostructure, and that between the ε F and the VBM via the projected band structures and the PDOS data (see Supplementary Fig. 2). The SM plot of the metal-contacted MoSi 2 N 4 and WSi 2 N 4 is shown in Fig. 5b-e for both electron-type and hole-type SBH. In general, the SM plot of the electron-type SBH exhibits a positive slope with increasing metal work function. Conversely, the hole-type SBH exhibits a negative slope as dictated by the SM relation in Eq. (2). A linear fit across the 11 metal contacts reveals a remarkably large slope parameter of S e ≈ S h = 0.69 for MoSi 2 N 4 where S e and S h denote electron and hole slope parameters, and S e = 0.77 and S h = 0.76 for WSi 2 N 4 . Although being lower than that of the 2D/2D and 3D/2D VDWtype contacts 17,18 , these S values-achieved intrinsically without relying on VDW-type contact engineering-are still significantly higher than that of almost all previously reported 2D semiconductors 14,15,27,[36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51] [Fig. 1e], such as MoS 2 monolayer and bilayer, WS 2 , InSe, black and blue phosphorene, arsenene, and silicene. Such exceptionally large S values, which spans over a large metal work function range of ΔW M = 2.7 eV (i.e., W M ranging from 3.3 to 6.0 eV), indicates greater flexibility in designing metal/MoSi 2 N 4 and metal/WSi 2 N 4 heterostructure with a specific SBH as required by the specific device applications. For completeness, the interface potential difference ΔV is also calculated and is presented in the Supplementary Notes and Supplementary Fig. 6.
Close contacts of Pd/MoGe 2 P 4 and Pd/WGe 2 P 4 To further understand whether the above described built-in atomic layer protection is also present in other sister 2D semiconducting monolayers in the MA 2 Z 4 family, such as the MoGe 2 P 4 and WGe 2 P 4 monolayers, we perform a close contact simulation of Pd/MoGe 2 P 4 and Pd/WGe 2 P 4 heterostructures. The close contact of Pd/MoGe 2 P 4 and Pd/MoGe 2 P 4 heterostructures are simulated by matching the 2 × 2 MoGe 2 P 4 (WGe 2 P 4 ) with the ffiffi ffi 7 p ffiffi ffi 7 p Pd(101) with a fixed interlayer separation of 2 Å. The lattice parameters are fixed to a/b = 3.54 Å for MoGe 2 P 4 and a/b = 3.55 Å for WGe 2 P 4 . The lattice structures are shown in Fig. 6a, d for Pd/MoGe 2 P 4 and Pd/WGe 2 P 4 , respectively. In the monolayer form, MoGe 2 P 4 and WGe 2 P 4 exhibit a band gap of 0.50 [ Fig. 6b] and 0.48 eV [Fig. 6e], respectively, which are significantly smaller than those of the MoSi 2 N 4 and WSi 2 N 4 monolayer. Importantly, when forming contact heterostructures with Pd, although some metalization and mid-gap states are observed in the band structures and the projected electronic DOS calculations due to the stronger interaction between the metal atoms and the outermost P atoms Fig. 6c, f for Pd/MoGe 2 P 4 and Pd/WGe 2 P 4 , respectively], it is remarkable to see that even with such a "close contact", the semiconducting bands at the K points remain intact. The weak FLP and the protection of the semiconducting electronic bands are thus expected to present in the broader family of (Mo, W)A 2 Z 4 with A = Si, Ge and Z = N, P. The physics of metal contacts to (Mo,W)Ge 2 N 4 and (Mo,W)Ge 2 P 4 shall form an interesting topic to explore in future works.
Metal contacts to bilayer MoSi 2 N 4 , MoGe 2 N 4 , and WGe 2 N 4 We further examine three examples of metal contacts to semiconducting monolayers from the MA 2 Z 4 family 58 beyond the MoSi 2 N 4 and WSi 2 N 4 monolayers, namely: (i) Au/ bilayer-MoSi 2 N 4 59 ; (ii) Au/MoGe 2 N 4 60 ; and (iii) Au/WGe 2 N 4 contacts. The lattice structure and the electronic band structure of Au/ bilayer-MoSi 2 N 4 contact are shown in Fig. 7a, b, respectively. The MoSi 2 N 4 /Au contact exhibits a convenient Ohmic behavior with minimal metal-semiconductor hybridization. For Au/MoGe 2 N 4 [ Fig. 7c-e], an Ohmic contact is also obtained while for Au/ WGe 2 N 4 [ Fig. 7f-h], the contact type exhibits an n-type Schottky contact characteristic. Importantly, the bands correspond to the semiconducting monolayers, particularly the electronic states around the VBM and CBM, remain well-intact in these contact heterostructures. These results thus suggest that the resistance against strong hybridization with the contacting metal appears to be a rather common feature among the septuple-layered semiconducting members of MA 2 Z 4 .

DISCUSSIONS
Experimentally, the S values are found to be lower than the theoretically calculated S values and such deviation is often associated with the presence of defects 61 . For example, the presence of S vacancies in metal contacts to MoS 2 has been theoretically and experimentally 62,63 shown to create defect states that can pin the Fermi level and hence reduces the S value. Firstprinciple simulations of defected metal contacts to MoSi 2 N 4 and WSi 2 N 4 monolayers shall quantitatively unravel how defects can affect the performance of metal contacts to these monolayers. Importantly, Sc and Ti contacts, both of which form excellent Ohmic contact with zero interfacial tunneling barrier to MoSi 2 N 4 and WSi 2 N 4 monolayers, are CMOS-compatible metals, thus revealing a practical contact engineering strategy for the construction of high-performance nanodevices based on MoSi 2 N 4 and WSi 2 N 4 . For 2D metal contacts, NbS 2 can be more beneficial than graphene in terms of charge injection efficiency due to the ultralow SBH 59 . Nonetheless, the identifications of 3D metal Ohmic contacts with zero SBH and zero interface tunneling barrier, such as Ti, clearly reveals the advantage of 3D metal contact over graphene and NbS 2 towards the design of high-efficiency electrical contact to MoSi 2 N 4 and WSi 2 N 4 monolayers.
Recently, the edge-type contact geometry has also received much research attention both 14,61 and experimentally [64][65][66][67] . Although the covalent-bonded heterointerface in edge-type can better facilitate charge injection, such contact geometry is faced with several challenges. The fabrication of high-quality edge-type contact remains challenging 1,68 , and the contact resistance of edge-type contact typically lies in the range of few tens of kΩcm 264-67 , which is one to two orders of magnitude higher than that of the face-type contact 19,69-71 (see Supplementary Table 4).
Recent work 72 has experimentally demonstrated that the edgetype contact may not be beneficial for contact application due to the presence of pronouncing electron scatterings by defects. In addition, the edge-type contact to MoS 2 is also found to be affected by a strong FLP effect (S = −0.09) 61 . As much interfacial physics of edge-type contact has yet to be unearthed, whether appropriately designed edge contacts to the MA 2 Z 4 monolayers could offer a route towards high-performance contacts remains an interesting open question to be addressed in future works. Finally, we note that the contact geometry can also take a combined edge-and face-type configuration. However, due to the large surface-to-contact ratio 1 , the face-type contact is still expected to play a more dominant role in determining the physical properties of such combined-type contact.
In conclusion, we show that the septuple-layered morphology and the electronic properties of MoSi 2 N 4 and WSi 2 N 4 monolayers offer a mechanism, alternative and complementary to the VDW contact paradigm, for suppressing the FLP effect and for achieving efficient tunable SBH in metal contacts to MoSi 2 N 4 and WSi 2 N 4 monolayers. The origin of such efficient protection of the semiconducting electronic states and the suppression of the FLP is a combination of three unusual behaviors of MoSi 2 N 4 and WSi 2 N 4 : (i) the septuple-layered structure which increases the atomic separation between the 2D semiconducting core layer and the external contacting metal atoms; (ii) the electronic structures in which the CBM and VBM states are concentrated in the inner core layer; and (iii) the relatively inert nature of the Si-N outermost layer. The DFT calculations presented here shall form a harbinger for the study of interfacial contact physics in the expansive family of MA 2 Z 4 . Myriads of phenomena, such as the evolution of SBH and FLP with a different number of layers 73 , the nature of 2D/2D and 2D/3D contacts for the other semiconducting members of the expansive MA 2 Z 4 family, the design of MA 2 Z 4 contact or heterostructures that can facilitate unusual non-charge transport, such as spin and valley transport 74 , or neuromorphic device operations 13 , remain to be explored. As the study of MA 2 Z 4 monolayer and few-layer family is still in its early infancy, the experimental synthesis of this material class shall represent one of the most important research quests to be tackled before the physics and the device application potential of this material family can be better understood. Future computational and experimental studies of electrical contacts to MA 2 Z 4 shall bring more surprises on the fundamental interface physics and chemistry, as well as the practical design of MA 2 Z 4 -based heterostructures and devices.   7 Metal contacts to several other species of MA 2 Z 4 monolayers. a, b Lattice structure, and the electronic band structure as well as the projected density of states of Au/bilayer-MoSi 2 N 4 contact, respectively. c-e shows the lattice structure, band structure of isolated monolayer, and the band structure of the contact heterostructure with the corresponding projected density of states of Au/MoGe 2 N 4 contact, respectively. f-h same as c-e but for Au/WGe 2 N 4 contact.

METHODS DFT calculations
The first-principle calculations are performed using the Vienna Ab initio Simulation Package (VASP) with projector augmented wave method [75][76][77][78] . The generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof realization (PBE) 79 is selected for the exchangecorrelation functional. A kinetic energy cut-off of 500 eV is set, and the Monkhorst-Pack k mesh 80 is used in the gamma-centered Brillouin zone with grids of 11 × 11 × 1 for the geometric optimizations and electronic calculations. Atomic geometry optimizations are terminated until all forces are smaller than 10 −3 eV/Å to ensure accuracy. The DFT-D3 Method 81 with the Grimme scheme is adopted. A vacuum space of 20 Å is set in the direction perpendicular to 3D-metal/2D-semiconductor surface. The calculations for Ni contacts are spin-polarized and the initial magnetic state for Ni substrate is set as ferromagnetic.
The 3D-metal/2D-semiconductor heterostructures are simulated by six layers of metal atoms, and two bottom layers of metals are fixed in atomic optimizations to simulate the internal atoms in the electrode. The lattice matching of the heterostructures are as followed: the 1 × 1 MoSi 2 N 4 (WSi 2 N 4 ) matches the 1 × 1 Ag(111)/Au(111)/Pd(111)/Pt(111)/Ti(0001), the ffiffi ffi 3 p ffiffi ffi 3 p MoSi 2 N 4 (WSi 2 N 4 ) matches the 2 × 2 Cu(111)/Ni(111)/Graphene, and the 2 × 2 MoSi 2 N 4 (WSi 2 N 4 ) matches the ffiffi ffi 3 p ffiffi ffi 3 p In(101)/Sc(0001)/ NbS 2 . The contact faces of the metals are chosen based on those employed in previous DFT studies due to their low surface free energies and thermodynamical stability 14,29,39 . The polycrystalline face is not considered in our calculations. To avoid the properties of MoSi 2 N 4 (WSi 2 N 4 ) being affected by mechanical stress when contacted by metals, we fixed the lattice parameters of semiconductors (MoSi 2 N 4 with a/b = 2.910 Å, and WSi 2 N 4 with a/b = 2.913 Å) and applied strains in the metals. The lattice mismatch for all heterostructures is less than 3.8%, and the calculation details of the metal-contacted MoSi 2 N 4 /WSi 2 N 4 monolayers are listed in Supplementary Tables 1 and 2.

DATA AVAILABILITY
The data of this study are available from the corresponding author upon reasonable request.